Two-loop Level Research Articles

The UV sensitivity of the Higgs potential in Gauge–Higgs Unification

Abstract In this paper, we discuss the UV sensitivity of the Higgs effective potential in a Gauge--Higgs Unification (GHU) model. We consider an $SU(\mathcal N)$ GHU on $\mathbf M^4\times S^1$ spacetime with a massless Dirac fermion. In this model, we evaluate the four-Fermi diagrams at the two-loop level and find them to be logarithmically divergent in the dimensional regularization scheme. Moreover, we confirm that their counter-terms contribute to the Higgs effective potential at the four-loop level. This result means that the Higgs effective potential in the GHU depends on UV theories as well as in other non-renormalizable theories.

Singlet-assisted electroweak phase transition at two loops

We investigate the electroweak phase transition in the real-singlet extension of the Standard Model at two-loop level, building upon existing one-loop studies. We calculate the effective potential in the high-temperature approximation and detail the required resummations at two-loop order. In typical strong-transition scenarios, we find deviations of order $20\% - 50\%$ from one-loop results in transition strength and critical temperature for both one- and two-step phase transitions. For extremely strong transitions, the discrepancy with one-loop predictions is even larger, presumably due to sizable scalar couplings in the tree-level potential. Along the way, we obtain a dimensionally-reduced effective theory applicable for non-perturbative lattice studies of the model.

Lessons fromTμμon inflation models: Two-loop renormalization ofηin the scalar QED

A non-minimal coupling $\eta$ has been attracting growing interest particularly in the context of inflation models, though its quantum nature is not clear yet. We study the renormalization of a non-minimal coupling in the scalar quantum electrodynamics (QED). We find ${\it no}$ inhomogeneous term of the renormalization group equation (RGE) at the two-loop level. This is similar to other theories, where an inhomogeneous term of the RGE appears only at a higher-loop order: e.g., four-loop order in $\lambda \phi^{4}$ theory.

A two loop induced neutrino mass model with modular A4 symmetry

We propose a model with radiatively induced neutrino mass at two-loop level, applying modular A4 symmetry. The neutrino mass matrix is formulated where the structure of associated couplings is restricted by the symmetry. Then we show several predictions in the lepton sector, satisfying lepton flavor violations as well as neutrino oscillation data. We also discuss muon anomalous magnetic moment and briefly comment on dark matter candidate.

Two-loop matching of renormalizable operators: general considerations and applications

Low-energy effective field theories (EFT) encode information about the physics at high energies — i.e., the high-energy theory (HET). To extract this information the EFT and the HET have to be matched to each other. At the one-loop level, general results for the matching of renormalizable operators have already been obtained in the literature. In the present paper, we take a step towards a better understanding of renormalizable operator matching at the two-loop level: focusing on the diagrammatic method, we discuss in detail the various contributions to two-loop matching conditions and compare different approaches to derive them. Moreover, we discuss which observables are best suited for the derivation of matching conditions. As a concrete application, we calculate the mathcal{O}left({alpha}_t{alpha}_sright) and mathcal{O}left({alpha}_t^2right) matching conditions of the scalar four-point couplings between the Standard Model (SM) and the Two-Higgs-Doublet Model (THDM) as well as the THDM and the Minimal Supersymmetric Standard Model (MSSM). We use the derived formulas to improve the prediction of the SM-like Higgs mass in the MSSM using the THDM as EFT.

CP violating hW+W\u2212 coupling in the Standard Model and beyond

Inspired by the recent development in determining the property of the observed Higgs boson, we explore the CP-violating (CPV) -{c}_{mathrm{CPV}}{hW}^{+mu nu}{tilde{W}}_{mu nu}^{-}/upsilon coupling in the Standard Model (SM) and beyond, where W±μν and {tilde{W}}^{pm mu nu} denote the W-boson field strength and its dual. To begin with, we show that the leading-order SM contribution to this CPV vertex appears at two-loop level. By summing over the quark flavor indices in the two loop integrals analytically, we can estimate the order of the corresponding Wilson coefficient to be {c}_{mathrm{CPV}}^{mathrm{SM}}sim mathcal{O}left({10}^{-23}right) , which is obviously too small to be probed at the LHC and planned future colliders. Then we investigate this CPV hW+W− interaction in two Beyond the Standard Model benchmark models: the left-right model and the complex 2-Higgs doublet model (C2HDM). Unlike what happens for the SM, the dominant contributions in both models arise at the one-loop level, and the corresponding Wilson coefficient can be as large as of mathcal{O} (10−9) in the former model and of mathcal{O} (10−3) for the latter. In light of such a large CPV effect in the hW+W− coupling, we also give the formulae for the leading one-loop contribution to the related CPV hZZ effective operator in the complex 2-Higgs doublet model. The order of magnitude of the Wilson coefficients in the C2HDM may be within reach of the high-luminosity LHC or planned future colliders.

Quasinormal modes of dirty black holes in the effective theory of gravity with a third order curvature term

We consider gravitational quasinormal modes of the static and spherically-symmetric dirty black holes in the effective theory of gravity which is renormalizable at the two-loop level. It is demonstrated that using the WKB-Pad\'e summation proposed in \cite{jaOp} one can achieve sufficient accuracy to calculate corrections to the complex frequencies of the quasinormal modes caused by the Goroff-Sagnotti curvature terms. It is shown that the Goroff-Sagnotti correction (with our choice of the sign of the coupling constant) increases damping of the fundamental modes (except for the lowest fundamental mode) and decreases their frequencies. We argue that the methods adopted in this paper can be used in the analysis of the influence of the higher-order curvature terms upon the quasinormal modes and in a number of related problems that require high accuracy.

A two-loop induced neutrino mass model, dark matter, and LFV processes ℓi → ℓjγ, and μe → ee in a hidden local U(1) symmetry

We discuss a model based on a hidden U(1)X gauge symmetry in which neutrino mass is induced at two-loop level by effects of interactions among particles in hidden sector and the Standard Model leptons. Since neutrino mass is suppressed by two-loop, its associate Yukawa couplings can be sizable and it would affect lepton flavor phenomenology. We analyze neutrino mass matrix, lepton flavor violating processes, electron/muon g−2 and dark matter annihilation cross section which are induced via interactions among Standard Model leptons and particles in U(1)X hidden sector, and their interactions can be sizable in our scenario. Performing numerical analysis, we show expected ratios for these processes using allowed parameters which can fit the neutrino data and satisfy flavor constraints.

Universality of ultra-relativistic gravitational scattering

We discuss the ultra-relativistic gravitational scattering of two massive particles at two-loop (3PM) level. We find that in this limit the real part of the eikonal, determining the deflection angle, is universal for gravitational theories in the two derivative approximation. This means that, regardless of the number of supersymmetries or the nature of the probes, the result connects smoothly with the massless case discussed since the late eighties by Amati, Ciafaloni and Veneziano. We analyse the problem both by using the analyticity and crossing properties of the scattering amplitudes and, in the case of the maximally supersymmetric theory, by explicit evaluation of the 4-point 2-loop amplitude using the results for the integrals in the full soft region. The first approach shows that the observable we are interested in is determined by the inelastic tree-level amplitude describing the emission of a graviton in the high-energy double-Regge limit, which is the origin of the universality property mentioned above. The second approach strongly suggests that the inclusion of the whole soft region is a necessary (and possibly sufficient) ingredient for recovering ultra relativistic finiteness and universality at the 3PM level. We conjecture that this universality persists at all orders in the PM expansion.

Dark Side of Weyl Gravity

We address the issue of a dynamical breakdown of scale invariance in quantum Weyl gravity together with related cosmological implications. In the first part, we build on our previous work [Phys. Rev. D2020, 101, 044050], where we found a non-trivial renormalization group fixed point in the infrared sector of quantum Weyl gravity. Here, we prove that the ensuing non-Gaussian IR fixed point is renormalization scheme independent. This confirms the feasibility of the analog of asymptotic safety scenario for quantum Weyl gravity in the IR. Some features, including non-analyticity and a lack of autonomy, of the system of β-functions near a turning point of the renormalization group at intermediate energies are also described. We further discuss an extension of the renormalization group analysis to the two-loop level. In particular, we show universal properties of the system of β-functions related to three couplings associated with C2 (Weyl square), G (Gauss–Bonnet), and R2 (Ricci curvature square) terms. Finally, we discuss various technical and conceptual issues associated with the conformal (trace) anomaly and propose possible remedies. In the second part, we analyze physics in the broken phase. In particular, we show that, in the low-energy sector of the broken phase, the theory looks like Starobinsky f(R) gravity with a gravi-cosmological constant that has a negative sign in comparison to the usual matter-induced cosmological constant. We discuss implications for cosmic inflation and highlight a non-trivial relation between Starobinsky’s parameter and the gravi-cosmological constant. Salient issues, including possible UV completions of quantum Weyl gravity and the role of the trace anomaly matching, are also discussed.

Practice of Constructing Nondegenerate Two-Loop Generalized IBP Identities

The problem of the systematic construction of differential operators for a generalized IBP reduction at a two-loop level is considered. The deeply optimized software makes it possible to efficiently construct such operators for the first nondegenerate two-loop cases. It is found that the most efficient approach is to construct the so-called partial operators that are much simpler than the complete operators and that change the power of only one polynomial.

Anomalous Dimensions of Leading Composite Operators in the Kinematic MHD Turbulence: Two-Loop Renormalization Group Analysis

Using the field theoretic renormalization group technique and the operator product expansion, the kinematic MHD turbulence is investigated in the second order (two-loop) approximation of the corresponding perturbative expansion. The anomalous dimensions of the leading composite operators, which drive the anomalous scaling of the single-time two-point correlation functions of the passive magnetic field, are calculated. It is shown that the two-loop corrections to these anomalous dimensions are significant and lead to the more anomalous (more negative) values of the total two-loop anomalous dimensions. It also means that the anomalous scaling at the two-loop level of approximation is much more pronounced in the present model of passive vector advection than in the analogous model of passive scalar quantity advected by the turbulent velocity field driven by the stochastic Navier–Stokes equation.

Sequentially loop suppressed fermion masses from a single discrete symmetry

We propose a systematic and renormalizable sequential loop suppression mechanism to generate the hierarchy of the Standard Model fermion masses from one discrete symmetry. The discrete symmetry is sequentially softly broken in order to generate one-loop level masses for the bottom, charm, tau and muon leptons and two-loop level masses for the lightest Standard Model charged fermions. The tiny masses for the light active neutrinos are produced from radiative type-I seesaw mechanism, where the Dirac mass terms are effectively generated at two-loop level.

Calculation of master integrals in terms of elliptic multiple polylogarithms

In modern quantum field theory, one of the most important tasks is the calculation of loop integrals. Loop integrals appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta. Even though this problem has already been in place since the mid-twentieth century, we not only do not understand how to calculate all classes of these integrals beyond one loop, we do not even know in what class of functions the answer is expressed. To partially solve this problem, different variations of new functions usually called elliptic multiple polylogarithms have been introduced in the last decade. In this paper, we explore the possibilities and limitations of this class of functions. As a practical example, we chose the processes associated with the physics of heavy quarkonium at the two-loop level.

Dispersive construction of two-loop P→πππ(P=K,η) amplitudes

We present and develop a general dispersive framework allowing us to construct representations of the amplitudes for the processes $P\pi\to\pi\pi$, $P=K,\eta$, valid at the two-loop level in the low-energy expansion. The construction proceeds through a two-step iteration, starting from the tree-level amplitudes and their S and P partial-wave projections. The one-loop amplitudes are obtained for all possible configurations of pion masses. The second iteration is presented in detail in the cases where either all masses of charged and neutral pions are equal, or for the decay into three neutral pions. Issues related to analyticity properties of the amplitudes and of their lowest partial-wave projections are given particular attention. This study is introduced by a brief survey of the situation, for both experimental and theoretical aspects, of the decay modes into three pions of charged and neutral kaons and of the eta meson.

A study of nucleon-nucleon scattering in covariant chiral effective field theory at the two-loop level

A study of nucleon-nucleon scattering in covariant chiral effective field theory at the two-loop level

Wilson-line scalar as a Nambu-Goldstone boson in flux compactifications and higher-loop corrections

We study a scalar zero mode originated from extradimensional components of a gauge field in a six-dimensional theory compactified on a magnetized torus. We confirm it is a Nambu-Goldstone boson of the translational symmetry on the torus which is breaking spontaneously due to magnetic flux. We also show explicitly it is massless up to the two-loop level. Moreover, we discuss full order contributions by considering the effective potential.

To be, or not to be finite? The Higgs potential in Gauge-Higgs Unification

In this paper, we investigate the finiteness of the Higgs effective potential in an SU( mathcal{N} ) Gauge-Higgs Unification (GHU) model defined on M4× S1. We obtain the Higgs effective potential at the two-loop level and find that it is finite. We also discuss that the Higgs effective potential is generically divergent for three- or higher-loop levels. As an example, we consider an SU( mathcal{N} ) gauge theory on M5× S1, where the one-loop corrections to the four-Fermi operators are divergent. We find that the Higgs effective potential depends on their counter terms at the three-loop level.

From gauged linear sigma models to geometric representation ofWCP(N,N˜)in 2D

In this paper two issues are addressed. First, we discuss renormalization properties of a class of gauged linear sigma models (GLSM) which reduce to $\mathbb{WCP}(N,\tilde{N})$ non-linear sigma models (NLSM) in the low-energy limit. Sometimes they are referred to as the Hanany-Tong models. If supersymmetry is ${\cal N} =(2,2)$ the ultraviolet-divergent logarithm in LGSM appears, in the renormalization of the Fayet-Iliopoulos parameter, and is exhausted by a single tadpole graph. This is not the case in the daughter NLSMs. As a result, the one-loop renormalizations are different in GLSMs and their daughter NLSMs We explain this difference and identify its source. In particular, we show why at $N=\tilde N$ there is no UV logarithms in the parent GLSM, while they do appear on the corresponding NLSM does not vanish. In the second part of the paper we discuss the same problem for a class of ${\cal N} =(0,2)$ GLSMs considered previously. In this case renormalization is not limited to one loop; all-orders exact $\beta$ functions for GLSMs are known. We discuss divergent loops at one and two-loop levels.

Wilson-Fisher fixed points for any dimension

The critical behavior of a non-local scalar field theory is studied. This theory has a non-local quartic interaction term which involves a real power -\beta of the Laplacian. The parameter \beta can be tuned so as to make that interaction marginal for any dimension. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization, and field renormalization are computed. In all cases a non-trivial IR fixed point is obtained. Remarkably, for dimensions different from 4, field renormalization is required at the one-loop level. For d=4, the theory reduces to the usual local \phi^{4} field theory and field renormalization is required starting at the the two-loop level. The critical exponents \nu and \eta are computed for dimensions 2,3,4 and 5. For dimensions greater than four, the critical exponent \eta turns out to be negative for \epsilon>0, which indicates a violation of the unitarity bounds.